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Braided Picard groups and graded extensions of braided tensor categories

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Abstract

We classify various types of graded extensions of a finite braided tensor category \(\mathcal {B}\) in terms of its 2-categorical Picard groups. In particular, we prove that braided extensions of \(\mathcal {B}\) by a finite group A correspond to braided monoidal 2-functors from A to the braided 2-categorical Picard group of \(\mathcal {B}\) (consisting of invertible central \(\mathcal {B}\)-module categories). Such functors can be expressed in terms of the Eilnberg-Mac Lane cohomology. We describe in detail braided 2-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.

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Notes

  1. For fusion categories, these 2-categories of module categories are fusion 2-categories [16].

  2. Below we omit the identity functors and the tensor product symbol \(\boxtimes \), so we write \(\mathcal {M}\mathcal {N}\) for \(\mathcal {M}\boxtimes \mathcal {N}\).

  3. Equalities of 2-cell compositions in this paper can be used to represent commuting polytopes [27]. These polytopes are recovered by gluing the diagrams on both sides of equality along the perimeter.

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Acknowledgements

We are grateful to Pavel Etingof, César Galindo, Corey Jones, David Jordan, Liang Kong, Victor Ostrik, and Milen Yakimov for many useful discussions. We also thank the anonymous referee for comments and corrections. The first author thanks the Simons foundation for partial support. The work of the second author was supported by the National Science Foundation under Grant No. DMS-1801198. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester.

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  1. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

    Alexei Davydov

  2. Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, USA

    Dmitri Nikshych

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  1. Alexei Davydov
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  2. Dmitri Nikshych
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Correspondence to Dmitri Nikshych.

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Davydov, A., Nikshych, D. Braided Picard groups and graded extensions of braided tensor categories. Sel. Math. New Ser. 27, 65 (2021). https://doi.org/10.1007/s00029-021-00670-1

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